CURVE SKETCHING
(Asymptotes)
rational function:
a function that can be expressed as 𝑓 𝑥 =
𝑝(𝑥)
𝑞(𝑥)
where:
 p(x) and q(x) are polynomial functions
 q(x)  0
Asymptotes are among the special features of rational
functions and play a significant role in curve sketching.
A rational function often has one or more discontinuities
which help define the shape of the function – these include
vertical asymptotes (infinite discontinuities).
The end behaviours of rational functions are determined by
either horizontal asymptotes or oblique asymptotes.
VERTICAL ASYMPTOTES
 𝑓 𝑥 =
𝑝(𝑥)
𝑞(𝑥)
has a vertical asymptote at x = c if q(c) = 0 and p(c) ≠ 0
1
Consider 𝑓 𝑥 = :
𝑥
y
x = 0 is a VA
lim 𝑓 𝑥 =
𝑥→0−
x
lim 𝑓 𝑥 =
𝑥→0+
VERTICAL ASYMPTOTES and INFINITE LIMITS
The graph of f(x) has a VA, x = c, if one of the following infinite limit
statements is true:
lim 𝑓 𝑥 = +∞ ,
𝑥→𝑐 −
lim 𝑓 𝑥 = −∞,
𝑥→𝑐 −
lim 𝑓(𝑥) = +∞,
𝑥→𝑐 +
lim 𝑓(𝑥) = −∞
𝑥→𝑐 +
HORIZONTAL ASYMPTOTES
 𝑓 𝑥 =
𝑝(𝑥)
𝑞(𝑥)
has a horizontal asymptote only when the degree of p(x) is
less than or equal to the degree of q(x)
1
Consider 𝑓 𝑥 = :
𝑥
y
y = 0 is a HA
lim 𝑓 𝑥 =
𝑥→∞
x
lim 𝑓 𝑥 =
𝑥→−∞
HORIZONTAL ASYMPTOTES and LIMITS at INFINITY
If lim 𝑓(𝑥) = 𝐿 or
𝑥→+∞
Ex.
lim 𝑓(𝑥) = 𝐿, then the line y = L is a HA of f(x).
𝑥→−∞
Determine the HA for each of the following:
a)
𝑓 𝑥 =
3𝑥+5
2𝑥−1
b)
𝑓 𝑥 =
4𝑥−1
2𝑥 2 +5𝑥−3
Can a function cross a HA?????
OBLIQUE ASYMPTOTES
 𝑓 𝑥 =
𝑝(𝑥)
𝑞(𝑥)
has an oblique asymptote only when the degree of p(x) is
exactly one greater than the degree of q(x)
Ex.
Determine the OA for 𝑓 𝑥 =
2𝑥 2 +3𝑥−1
𝑥+1
.
PUTTING IT ALL TOGETHER
Example 
a)
Determine all critical points, check for discontinuities, and sketch the
function in each of the following:
𝑓 𝑥 =
1
y
𝑥 2 −1
x
f ‘(x)
b)
𝑓 𝑥 =
𝑥
y
𝑥 2 −𝑥−6
x
f ‘(x)
c)
𝑓 𝑥 =
2𝑥 2 +𝑥−1
y
𝑥−1
x
f ‘(x)
p.193–195 #1, 2, 3cd, 6bd, 7ab, 10, 11, 15